Being quite new to the world of PDEs, I would like your help regarding a specific change of variable. Namely, I consider the inviscid Burgers equation :
\begin{equation}
u_t+uu_x=0
\end{equation}
And I would like to do the change of variable $(t,x) \mapsto (t,s(t,x))$ where :
\begin{equation}
\quad ds^2=dx^2+du^2 \quad \text{or} \quad ds = \sqrt{1+u_x^2}dx
\end{equation}
Using the chain rule, I have :
\begin{equation}
\begin{cases}
u_t = \frac{du}{dt} + \frac{du}{ds} \frac{ds}{dt} \\
u_x = \frac{du}{dt} \frac{dt}{dx} + \frac{du}{ds}\frac{ds}{dx} = u_s \sqrt{1+u_x^2} = \frac{u_s}{\sqrt{1-u_s^2}}
\end{cases}
\end{equation}
If I then replace into the PDE, I obtain :
\begin{equation}
u_t + u_s s_t + \frac{[u^2]_s}{2 \sqrt{1-u_s^2}} = 0
\end{equation}
This is where I request your help. I would like to know if the computations above make sense (provided that $u_s^2$ is never equals to 1), and how can I handle the term $s_t$ since I do not know its explicit expression ?
Moreover, if I want to do the reverse change of variable to get $u(t,x)$, how do I proceed ?